openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /SamplerComposition.lean
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalFrozenTablesLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.BucketUniformLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.SamplerComposition | |
| open PointwiseSpaces RecursiveSpaces DescendantSpaces RecursiveSampler | |
| noncomputable section | |
| universe u w | |
| variable {branch : Nat → Nat} {n m k : Nat} | |
| def ExpansionSpace (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) : | |
| {n m k : Nat} → (p : Path branch n m) → (q : Path branch m k) → | |
| (A : Slots branch n → Type u) → DrawIndex repeats p → Type (max u w) | |
| | _, _, _, .refl _, q, A, _ => RecursiveSampler.Tape 𝕜 repeats q A | |
| | n + 1, _, _, .step i p, q, A, j => | |
| match j with | |
| | Sum.inl j => squareSpace (space 𝕜 branch n (childFamily A j.val)) | |
| | Sum.inr j => ExpansionSpace 𝕜 repeats p q (childFamily A i) j.2 | |
| abbrev ExpandedTape (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) (q : Path branch m k) (A : Slots branch n → Type u) := | |
| (j : DrawIndex repeats p) → ExpansionSpace 𝕜 repeats p q A j | |
| def zeroFactor (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) : | |
| {n m k : Nat} → (p : Path branch n m) → (q : Path branch m k) → | |
| (A : Slots branch n → Type u) → (j : DrawIndex repeats p) → | |
| ExpansionSpace 𝕜 repeats p q A j | |
| | _, _, _, .refl _, q, A, _ => zeroTape 𝕜 repeats q A | |
| | n + 1, _, _, .step i p, q, A, j => | |
| match j with | |
| | Sum.inl j => (0 : squareSpace (space 𝕜 branch n (childFamily A j.val))) | |
| | Sum.inr j => zeroFactor 𝕜 repeats p q (childFamily A i) j.2 | |
| def zeroExpandedTape (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) (q : Path branch m k) (A : Slots branch n → Type u) : | |
| ExpandedTape 𝕜 repeats p q A := zeroFactor 𝕜 repeats p q A | |
| def expandedSubTape (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (i : Fin (branch n)) (p : Path branch n m) (q : Path branch m k) | |
| (A : Slots branch (n + 1) → Type u) | |
| (t : ExpandedTape 𝕜 repeats (.step i p) q A) | |
| (h : Fin (repeats (n + 1))) (b : Bool) : | |
| ExpandedTape 𝕜 repeats p q (childFamily A i) := fun j => t (.inr ((h, b), j)) | |
| def expand (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) : | |
| {n m k : Nat} → (p : Path branch n m) → (q : Path branch m k) → | |
| (A : Slots branch n → Type u) → RecursiveSampler.Tape 𝕜 repeats (p.append q) A → | |
| ExpandedTape 𝕜 repeats p q A | |
| | _, _, _, .refl _, _, _, t => fun _ => t | |
| | _n + 1, _, _, .step i p, q, A, t => fun j => | |
| match j with | |
| | Sum.inl j => t (.inl j) | |
| | Sum.inr j => expand 𝕜 repeats p q (childFamily A i) | |
| (subTape 𝕜 repeats i (p.append q) A t j.1.1 j.1.2) j.2 | |
| def flatten (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) : | |
| {n m k : Nat} → (p : Path branch n m) → (q : Path branch m k) → | |
| (A : Slots branch n → Type u) → ExpandedTape 𝕜 repeats p q A → | |
| RecursiveSampler.Tape 𝕜 repeats (p.append q) A | |
| | _, _, _, .refl _, _, _, t => t () | |
| | _n + 1, _, _, .step i p, q, A, t => fun j => | |
| match j with | |
| | Sum.inl j => t (.inl j) | |
| | Sum.inr j => flatten 𝕜 repeats p q (childFamily A i) | |
| (expandedSubTape 𝕜 repeats i p q A t j.1.1 j.1.2) j.2 | |
| theorem flatten_expand (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) : | |
| ∀ (q : Path branch m k) (A : Slots branch n → Type u) | |
| (t : RecursiveSampler.Tape 𝕜 repeats (p.append q) A), | |
| flatten 𝕜 repeats p q A (expand 𝕜 repeats p q A t) = t := by | |
| induction p with | |
| | refl n => intro q A t; rfl | |
| | @step n m i p ih => | |
| intro q A t | |
| funext j | |
| cases j with | |
| | inl j => rfl | |
| | inr j => | |
| exact congrFun | |
| (ih q (childFamily A i) (subTape 𝕜 repeats i (p.append q) A t j.1.1 j.1.2)) j.2 | |
| theorem expand_flatten (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) : | |
| ∀ (q : Path branch m k) (A : Slots branch n → Type u) | |
| (t : ExpandedTape 𝕜 repeats p q A), | |
| expand 𝕜 repeats p q A (flatten 𝕜 repeats p q A t) = t := by | |
| induction p with | |
| | refl n => | |
| intro q A t | |
| funext j | |
| cases j | |
| rfl | |
| | @step n m i p ih => | |
| intro q A t | |
| funext j | |
| cases j with | |
| | inl j => rfl | |
| | inr j => | |
| exact congrFun | |
| (ih q (childFamily A i) (expandedSubTape 𝕜 repeats i p q A t j.1.1 j.1.2)) j.2 | |
| def expandEquiv (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) (q : Path branch m k) (A : Slots branch n → Type u) : | |
| RecursiveSampler.Tape 𝕜 repeats (p.append q) A ≃ ExpandedTape 𝕜 repeats p q A where | |
| toFun := expand 𝕜 repeats p q A | |
| invFun := flatten 𝕜 repeats p q A | |
| left_inv := flatten_expand 𝕜 repeats p q A | |
| right_inv := expand_flatten 𝕜 repeats p q A | |
| def collapseFactor (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) : | |
| {n m k : Nat} → (p : Path branch n m) → (q : Path branch m k) → | |
| (A : Slots branch n → Type u) → (j : DrawIndex repeats p) → | |
| ExpansionSpace 𝕜 repeats p q A j → DrawSpace 𝕜 repeats p A j | |
| | _, _, _, .refl _, q, A, _ => RecursiveSampler.evaluate 𝕜 repeats q A | |
| | _, _, _, .step i p, q, A, j => | |
| match j with | |
| | Sum.inl _ => fun t => t | |
| | Sum.inr j => collapseFactor 𝕜 repeats p q (childFamily A i) j.2 | |
| def collapseExpanded (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) (q : Path branch m k) (A : Slots branch n → Type u) : | |
| ExpandedTape 𝕜 repeats p q A → RecursiveSampler.Tape 𝕜 repeats p A := | |
| fun t j => collapseFactor 𝕜 repeats p q A j (t j) | |
| def collapseSuffix (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) (q : Path branch m k) (A : Slots branch n → Type u) : | |
| RecursiveSampler.Tape 𝕜 repeats (p.append q) A → RecursiveSampler.Tape 𝕜 repeats p A := | |
| collapseExpanded 𝕜 repeats p q A ∘ expandEquiv 𝕜 repeats p q A | |
| theorem evaluate_collapseExpanded (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) : | |
| ∀ (q : Path branch m k) (A : Slots branch n → Type u) | |
| (t : ExpandedTape 𝕜 repeats p q A), | |
| RecursiveSampler.evaluate 𝕜 repeats p A (collapseExpanded 𝕜 repeats p q A t) = | |
| RecursiveSampler.evaluate 𝕜 repeats (p.append q) A (flatten 𝕜 repeats p q A t) := by | |
| induction p with | |
| | refl n => intro q A t; rfl | |
| | @step n m i p ih => | |
| intro q A t | |
| have hl := funext fun h : Fin (repeats (n + 1)) => | |
| ih q (childFamily A i) (expandedSubTape 𝕜 repeats i p q A t h false) | |
| have hr := funext fun h : Fin (repeats (n + 1)) => | |
| ih q (childFamily A i) (expandedSubTape 𝕜 repeats i p q A t h true) | |
| exact congrArg₂ | |
| (fun (left right : Fin (repeats (n + 1)) → space 𝕜 branch n (childFamily A i)) => | |
| combine 𝕜 A i (repeats (n + 1)) (fun j => t (.inl j)) left right) hl hr | |
| theorem evaluate_collapseSuffix (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (p : Path branch n m) (q : Path branch m k) (A : Slots branch n → Type u) | |
| (t : RecursiveSampler.Tape 𝕜 repeats (p.append q) A) : | |
| RecursiveSampler.evaluate 𝕜 repeats p A (collapseSuffix 𝕜 repeats p q A t) = | |
| RecursiveSampler.evaluate 𝕜 repeats (p.append q) A t := | |
| (evaluate_collapseExpanded 𝕜 repeats p q A (expand 𝕜 repeats p q A t)).trans | |
| (congrArg (RecursiveSampler.evaluate 𝕜 repeats (p.append q) A) | |
| (flatten_expand 𝕜 repeats p q A t)) | |
| end | |
| end PerfectCompleteness.SamplerComposition | |
| end | |
| end OAI | |